any nonzero integer is quadratic residue

Proof.1∘.a=2. We see that 32≡2(mod7) and 7∤2, whence 2 is a quadratic residue modulo 7.
2∘.2∣a but a≠2. The number 12-a=1-a (which is odd and ≠±1) has an odd prime factor p which does not divide a. Thus a is a quadratic residue modulo p.
3∘.a=3. We state that 42-3=13≡0(mod13) and 13∤3. Therefore 3 is a quadratic residue modulo 13.
4∘.a=5. We see that 42-5=11≡0(mod11) and 11∤5, i.e. 5 is a quadratic residue modulo 11.
5∘.2∤a but a≠3, a≠5. Now the number 22-a=4-a (which is odd and ≠±1) has an odd prime factor p. Moreover, p∤a since p∤4. Accordingly, a is a quadratic residue modulo p.